People put conditions on donations all the time. They donate to the Red Cross for a specific disaster. The donate money to a church for the building fund. They donate money to a hospital to buy a new x-ray machine. They donate money to the scouts for a new dining hall.
It is possible to donate money to a non-profit for a specific purpose. If the non-profit doesn't want to accept the money with those strings they can refuse. Generally these specific projects are initiated by the non-profit. But there is no requirement that the idea originate with the non-profit.
It is also up to the non-profit and their legal advisers regarding how strictly they view those strings. If you donate money for web design and they don't spend it all, can they pay net years hosting bill with the money, or must they hold it for a few years for when they need a designer again?
If the company wants to provide the service, they can structure the project to pay their employees for their time. They pay employees for $100 of labor while deigning the website. The pay and benefits reduce profit thus lowering taxes.
Donating money to the non-profit to be given back to the company doesn't seem to be the best way to structure transaction. At best it is a wash.
Donating money to a charity and then directing exactly which contractor will perform the service starts to look like money laundering, and most charities will get wary.